相关论文: The Kepler conjecture
Using experiments and simulations, we investigate the clusters that form when colloidal spheres stick irreversibly to -- or "park" on -- smaller spheres. We use either oppositely charged particles or particles labeled with complementary DNA…
In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict…
Astrophysical observations indicate that the ``Local Universe" has a relatively lower matter density ($\Omega_0$) than the predictions of the standard inflation cosmology and the large-scale motions of galaxies which provide a mean mass…
Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers…
As part of a stellar population sampling program, a series of photometric probes at various field sizes and depths have been obtained in a low extinction window in the galactic anticentre direction. Such data set strong constraints on the…
A celebrated result of Beck shows that for any set of $N$ points on $\mathbb{S}^d$ there always exists a spherical cap $B \subset \mathbb{S}^d$ such that number of points in the cap deviates from the expected value $\sigma(B) \cdot N$ by at…
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of…
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity.…
The method, proposed in \cite{Za22} to derive the densest packing fraction of random disc and sphere packings, is shown to yield in two dimensions too high a value that (i) violates the very assumption underlying the method and (ii)…
A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density…
We compile observations of the surface mass density profiles of dense stellar systems, including globular clusters in the Milky Way and nearby galaxies, massive star clusters in nearby starbursts, nuclear star clusters in dwarf spheroidals…
Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error…
The dark matter content of globular clusters, highly compact gravity-bound stellar systems, is unknown. It is also generally unknow*able*, due to their mass-to-light ratios typically ranging between 1$-$3 in solar units, accommodating a…
We investigate the density of compactly supported smooth functions in the Sobolev space $W^{k,p}$ on complete Riemannian manifolds. In the first part of the paper, we extend to the full range $p\in [1,2]$ the most general results known in…
The sintering behavior of close packed spheres is investigated using a numerical model. The investigated systems are the body centered cubic (BCC), face centered cubic (FCC) and hexagonal closed packed spheres (HCP). The sintering behavior…
Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the…
In the third Galactic quadrant (180 < l < 270) of the Milky Way, the Galactic thin disk exhibits a significant warp ---shown both by gas and young stars--- bending down a few kpc below the formal Galactic plane (b=0). This warp shows its…
The Cohn-Elkies linear programming (LP) bound for sphere packing is known to be sharp in dimensions 8 and 24 but in no other dimension above 2. We investigate why by examining three independent necessary conditions for LP sharpness, drawn…
In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower…
In this review, the status of measurements of the matter density (Omega), the vaccuum energy density or cosmological constant (Lambda), the Hubble constant (H0), and ages of the oldest measured objects (t0) are summarized. Measurements of…